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          <h1 class="post-title" itemprop="name headline">量化投资学习笔记49——通过问题学算法:04N皇后问题(Keep Those Queens Apart)</h1>
        

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        <p>《programming for the puzzled》第四章<br>涉及到的问题：二维列表，while循环，continue语句，函数的默认参数，用递归进行搜索，剪枝(Conflict detection)。<br>就是所谓的“八皇后问题”了，在一个8×8的国际象棋棋盘上放8个皇后，要求它们不能彼此攻击，即：<br>①没有两个皇后在同一行。<br>②没有两个皇后在同一列。<br>③没有两个皇后在同一对角线上。<br>先考虑小规模的问题，5×5的棋盘上放5个皇后。<br>我先自己尝试一下，用穷举吧。</p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment"># 穷举法</span></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">QJ</span>(<span class="params">n</span>):</span></span><br><span class="line">    board = [<span class="number">0</span>]*n</span><br><span class="line">    displayBoard(board)</span><br><span class="line">    <span class="comment"># 穷举每一种情况</span></span><br><span class="line">    flag = <span class="literal">False</span></span><br><span class="line">    </span><br><span class="line">    ans = []</span><br><span class="line">    count = <span class="number">0</span></span><br><span class="line">    times = <span class="number">0</span></span><br><span class="line">    </span><br><span class="line">    <span class="keyword">for</span> num <span class="keyword">in</span> <span class="built_in">range</span>(n**n):</span><br><span class="line">        temp = num</span><br><span class="line">        <span class="keyword">for</span> i <span class="keyword">in</span> <span class="built_in">range</span>(n):</span><br><span class="line">            board[i] = <span class="built_in">int</span>(temp % n)</span><br><span class="line">            temp = temp / n</span><br><span class="line">        flag = <span class="literal">True</span></span><br><span class="line">        </span><br><span class="line">        <span class="keyword">for</span> i <span class="keyword">in</span> <span class="built_in">range</span>(n):</span><br><span class="line">            <span class="keyword">for</span> j <span class="keyword">in</span> <span class="built_in">range</span>(i+<span class="number">1</span>, n):</span><br><span class="line">                <span class="keyword">if</span> (board[i] == board[j] <span class="keyword">or</span> <span class="built_in">abs</span>(i-j) == <span class="built_in">abs</span>(board[i] - board[j])):</span><br><span class="line">                    flag = <span class="literal">False</span></span><br><span class="line">                    <span class="keyword">break</span></span><br><span class="line">        <span class="keyword">if</span> flag == <span class="literal">True</span>:</span><br><span class="line">            count += <span class="number">1</span></span><br><span class="line">        times += <span class="number">1</span></span><br><span class="line">    <span class="keyword">return</span> count, times</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="keyword">if</span> __name__ == <span class="string">&quot;__main__&quot;</span>:</span><br><span class="line">    n = <span class="built_in">int</span>(<span class="built_in">input</span>(<span class="string">&quot;输入问题规模:(&gt;0)&quot;</span>))</span><br><span class="line">    count, times = QJ(n)</span><br><span class="line">    print(<span class="string">&quot;答案总数:&quot;</span>, count, <span class="string">&quot;,运算次数:&quot;</span>, times)</span><br></pre></td></tr></table></figure>
<p>还是照网上的抄的，用一维列表表示棋盘上每行放皇后的那列的位置。<br>再来看书里的解法，用的是二维列表表示棋盘。<br>关键是判断一个放法是否符合问题的三个要求。</p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment"># 工具函数，检查棋子布局是否合规</span></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">noConflicts</span>(<span class="params">board, current, qindex, n</span>):</span></span><br><span class="line">    <span class="keyword">for</span> j <span class="keyword">in</span> <span class="built_in">range</span>(current):</span><br><span class="line">        <span class="keyword">if</span> board[qindex][j] == <span class="number">1</span>:</span><br><span class="line">            <span class="keyword">return</span> <span class="literal">False</span></span><br><span class="line">            </span><br><span class="line">    k = <span class="number">1</span></span><br><span class="line">    <span class="keyword">while</span> qindex - k &gt;= <span class="number">0</span> <span class="keyword">and</span> current - k &gt;= <span class="number">0</span>:</span><br><span class="line">        <span class="keyword">if</span> board[qindex - k][current - k] == <span class="number">1</span>:</span><br><span class="line">            <span class="keyword">return</span> <span class="literal">False</span></span><br><span class="line">        k += <span class="number">1</span></span><br><span class="line">    k = <span class="number">1</span></span><br><span class="line">    <span class="keyword">while</span> qindex + k &lt; n <span class="keyword">and</span> current - k &gt;= <span class="number">0</span>:</span><br><span class="line">        <span class="keyword">if</span> board[qindex + k][current - k] == <span class="number">1</span>:</span><br><span class="line">            <span class="keyword">return</span> <span class="literal">False</span></span><br><span class="line">        k += <span class="number">1</span></span><br><span class="line">    </span><br><span class="line">    <span class="keyword">return</span> <span class="literal">True</span></span><br></pre></td></tr></table></figure>
<p>qindex是已经放了棋子的行的位置，相应的列的位置是current。<br>我自己照着书敲，老是出错，结果不对。把书的代码复制粘贴过来运行才发现是棋盘初始化出错了。</p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br><span class="line">37</span><br><span class="line">38</span><br><span class="line">39</span><br><span class="line">40</span><br><span class="line">41</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment"># 工具函数 输出二维棋盘</span></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">displayBoard</span>(<span class="params">board</span>):</span></span><br><span class="line">    n = <span class="built_in">len</span>(board)</span><br><span class="line">    <span class="keyword">for</span> i <span class="keyword">in</span> <span class="built_in">range</span>(n):</span><br><span class="line">        print(board[i])</span><br><span class="line">    print(<span class="string">&quot;\n&quot;</span>)</span><br><span class="line"></span><br><span class="line"></span><br><span class="line"><span class="comment">#This procedure places 4 Queens on a board so they don&#x27;t conflict</span></span><br><span class="line"><span class="comment">#It assumes n = 4 and won&#x27;t work with other n!</span></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">FourQueens</span>(<span class="params">n=<span class="number">4</span></span>):</span></span><br><span class="line">    <span class="comment">#Initialize the board to be empty</span></span><br><span class="line">    board = [ [<span class="number">0</span>, <span class="number">0</span>, <span class="number">0</span>, <span class="number">0</span>],</span><br><span class="line">              [<span class="number">0</span>, <span class="number">0</span>, <span class="number">0</span>, <span class="number">0</span>],</span><br><span class="line">              [<span class="number">0</span>, <span class="number">0</span>, <span class="number">0</span>, <span class="number">0</span>],</span><br><span class="line">              [<span class="number">0</span>, <span class="number">0</span>, <span class="number">0</span>, <span class="number">0</span>] ]</span><br><span class="line">    <span class="comment"># 照下面两行初始化会出错</span></span><br><span class="line">    <span class="comment"># board = [[0]*n]*n</span></span><br><span class="line">    <span class="comment"># print(len(board), len(board[0]))</span></span><br><span class="line">   </span><br><span class="line">    <span class="comment">#Place a queen a column at a time beginning with leftmost column</span></span><br><span class="line">    count, times = <span class="number">0</span>, <span class="number">0</span></span><br><span class="line">    <span class="keyword">for</span> i <span class="keyword">in</span> <span class="built_in">range</span>(n):</span><br><span class="line">        board[i][<span class="number">0</span>] = <span class="number">1</span></span><br><span class="line">        <span class="keyword">for</span> j <span class="keyword">in</span> <span class="built_in">range</span>(n):</span><br><span class="line">            board[j][<span class="number">1</span>] = <span class="number">1</span></span><br><span class="line">            <span class="keyword">if</span> noConflicts(board, <span class="number">1</span>, j, n):</span><br><span class="line">                <span class="keyword">for</span> k <span class="keyword">in</span> <span class="built_in">range</span>(n):</span><br><span class="line">                    board[k][<span class="number">2</span>] = <span class="number">1</span></span><br><span class="line">                    <span class="keyword">if</span> noConflicts(board, <span class="number">2</span>, k, n):</span><br><span class="line">                        <span class="keyword">for</span> m <span class="keyword">in</span> <span class="built_in">range</span>(n):</span><br><span class="line">                            board[m][<span class="number">3</span>] = <span class="number">1</span></span><br><span class="line">                            <span class="keyword">if</span> noConflicts(board, <span class="number">3</span>, m, n):</span><br><span class="line">                                displayBoard(board)</span><br><span class="line">                                count += <span class="number">1</span></span><br><span class="line">                            times += <span class="number">1</span></span><br><span class="line">                            board[m][<span class="number">3</span>] = <span class="number">0</span></span><br><span class="line">                    board[k][<span class="number">2</span>] = <span class="number">0</span></span><br><span class="line">            board[j][<span class="number">1</span>] = <span class="number">0</span></span><br><span class="line">        board[i][<span class="number">0</span>] = <span class="number">0</span></span><br><span class="line">    <span class="keyword">return</span> count, times</span><br></pre></td></tr></table></figure>
<p>2维数组代表棋盘是一个很自然的想法，但由于我们在每一列仅放一个皇后（每一行也是一样的），我们可以用一维数组代表某一列放皇后的位置，而其取值代表了在该列所放皇后的行数。<br>每个元素的取值范围为[-1,3]，-1意味着这一列没有放棋子，0代表放在该列的第一行，3代表放在该列的最后一行。</p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment"># 判断布局是否违反规则   </span></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">noConflicts2</span>(<span class="params">board, current</span>):</span></span><br><span class="line">    <span class="keyword">for</span> i <span class="keyword">in</span> <span class="built_in">range</span>(current):</span><br><span class="line">        <span class="comment"># 每行一个</span></span><br><span class="line">        <span class="keyword">if</span> (board[i] == board[current]):</span><br><span class="line">            <span class="keyword">return</span> <span class="literal">False</span></span><br><span class="line">        <span class="comment"># 对角线</span></span><br><span class="line">        <span class="keyword">if</span> (current - i == <span class="built_in">abs</span>(board[current] - board[i])):</span><br><span class="line">            <span class="keyword">return</span> <span class="literal">False</span></span><br><span class="line">    <span class="keyword">return</span> <span class="literal">True</span></span><br></pre></td></tr></table></figure>
<p>现在来做八皇后问题。</p>
<figure class="highlight python"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span class="line">25</span><br><span class="line">26</span><br><span class="line">27</span><br><span class="line">28</span><br><span class="line">29</span><br><span class="line">30</span><br><span class="line">31</span><br><span class="line">32</span><br><span class="line">33</span><br><span class="line">34</span><br><span class="line">35</span><br><span class="line">36</span><br></pre></td><td class="code"><pre><span class="line"><span class="comment"># 八皇后 回溯</span></span><br><span class="line"><span class="function"><span class="keyword">def</span> <span class="title">EightQueens</span>(<span class="params">n=<span class="number">8</span></span>):</span></span><br><span class="line">    board = [-<span class="number">1</span>]*n</span><br><span class="line">    count = <span class="number">0</span></span><br><span class="line">    <span class="keyword">for</span> i <span class="keyword">in</span> <span class="built_in">range</span>(n):</span><br><span class="line">        board[<span class="number">0</span>] = i</span><br><span class="line">        <span class="keyword">for</span> j <span class="keyword">in</span> <span class="built_in">range</span>(n):</span><br><span class="line">            board[<span class="number">1</span>] = j</span><br><span class="line">            <span class="keyword">if</span> <span class="keyword">not</span> noConflicts2(board, <span class="number">1</span>):</span><br><span class="line">                <span class="keyword">continue</span></span><br><span class="line">            <span class="keyword">for</span> k <span class="keyword">in</span> <span class="built_in">range</span>(n):</span><br><span class="line">                board[<span class="number">2</span>] = k</span><br><span class="line">                <span class="keyword">if</span> <span class="keyword">not</span> noConflicts2(board, <span class="number">2</span>):</span><br><span class="line">                    <span class="keyword">continue</span></span><br><span class="line">                <span class="keyword">for</span> l <span class="keyword">in</span> <span class="built_in">range</span>(n):</span><br><span class="line">                    board[<span class="number">3</span>] = l</span><br><span class="line">                    <span class="keyword">if</span> <span class="keyword">not</span> noConflicts2(board, <span class="number">3</span>):</span><br><span class="line">                        <span class="keyword">continue</span></span><br><span class="line">                    <span class="keyword">for</span> m <span class="keyword">in</span> <span class="built_in">range</span>(n):</span><br><span class="line">                        board[<span class="number">4</span>] = m</span><br><span class="line">                        <span class="keyword">if</span> <span class="keyword">not</span> noConflicts2(board, <span class="number">4</span>):</span><br><span class="line">                            <span class="keyword">continue</span></span><br><span class="line">                        <span class="keyword">for</span> o <span class="keyword">in</span> <span class="built_in">range</span>(n):</span><br><span class="line">                            board[<span class="number">5</span>] = o</span><br><span class="line">                            <span class="keyword">if</span> <span class="keyword">not</span> noConflicts2(board, <span class="number">5</span>):</span><br><span class="line">                                <span class="keyword">continue</span></span><br><span class="line">                            <span class="keyword">for</span> p <span class="keyword">in</span> <span class="built_in">range</span>(n):</span><br><span class="line">                                board[<span class="number">6</span>] = p</span><br><span class="line">                                <span class="keyword">if</span> <span class="keyword">not</span> noConflicts2(board, <span class="number">6</span>):</span><br><span class="line">                                    <span class="keyword">continue</span></span><br><span class="line">                                <span class="keyword">for</span> q <span class="keyword">in</span> <span class="built_in">range</span>(n):</span><br><span class="line">                                    board[<span class="number">7</span>] = q</span><br><span class="line">                                    <span class="keyword">if</span> noConflicts2(board, <span class="number">7</span>):</span><br><span class="line">                                        print(board)</span><br><span class="line">                                        count += <span class="number">1</span></span><br><span class="line">    <span class="keyword">return</span> count</span><br></pre></td></tr></table></figure>
<p>跟前面穷举法相比，一旦遇到有冲突的落子方案，直接就pass了，快很多。<br>每次落子，都要调用监测是否有冲突的判断函数。<br>这是全书中“最丑”的代码了。可以改进的。用递归代码，在问题10的时候。<br>算法的核心思想是遍历所有可能的下棋方法，不遗漏任何一个可能。<br>另一种方法是在将布局完成后才检测其是否违反规则，而不是在下棋的过程中就检测。这种方法就是真正的穷举了，效率非常低。时间复杂度是指数级的O(n^n)。<br>练习1.增加一个变量，记录解的个数。我已经在上面做了。<br>练习2.修改代码，传入一个一维列表的下棋方式，如[-1,4,-1,-1,-1,-1,-1,0]，有两个位置已经下了棋了，输出可能的下棋方式。<br>增加一个判断，等于-1则按原来的赋值，如果不等于-1就跳过赋值，下下一列。但老也调不对。这种八个循环在一起是比较”丑”，过了。<br>本章代码： <a target="_blank" rel="noopener" href="https://github.com/zwdnet/MyQuant/blob/master/44/04">https://github.com/zwdnet/MyQuant/blob/master/44/04</a></p>
<p>我发文章的三个地方，欢迎大家在朋友圈等地方分享，欢迎点“在看”。<br>我的个人博客地址：<a href="https://zwdnet.github.io/">https://zwdnet.github.io</a><br>我的知乎文章地址： <a target="_blank" rel="noopener" href="https://www.zhihu.com/people/zhao-you-min/posts">https://www.zhihu.com/people/zhao-you-min/posts</a><br>我的微信个人订阅号：赵瑜敏的口腔医学学习园地</p>
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